MathDB
Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
2020 China National Olympiad
6
6
Part of
2020 China National Olympiad
Problems
(1)
Swapping coefficients gives roots
Source: 2020 CMO P6
11/27/2019
Does there exist positive reals
a
0
,
a
1
,
…
,
a
19
a_0, a_1,\ldots ,a_{19}
a
0
,
a
1
,
…
,
a
19
, such that the polynomial
P
(
x
)
=
x
20
+
a
19
x
19
+
…
+
a
1
x
+
a
0
P(x)=x^{20}+a_{19}x^{19}+\ldots +a_1x+a_0
P
(
x
)
=
x
20
+
a
19
x
19
+
…
+
a
1
x
+
a
0
does not have any real roots, yet all polynomials formed from swapping any two coefficients
a
i
,
a
j
a_i,a_j
a
i
,
a
j
has at least one real root?
algebra
polynomial